Minor-Closed Graph Classes with Bounded Layered Pathwidth Vida Dujmovi´c z David Eppstein y Gwena¨elJoret x Pat Morin ∗ David R. Wood { 19th October 2018; revised 4th June 2020 Abstract We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class. 1 Introduction Pathwidth and treewidth are graph parameters that respectively measure how similar a given graph is to a path or a tree. These parameters are of fundamental importance in structural graph theory, especially in Roberston and Seymour's graph minors series. They also have numerous applications in algorithmic graph theory. Indeed, many NP-complete problems are solvable in polynomial time on graphs of bounded treewidth [23]. Recently, Dujmovi´c,Morin, and Wood [19] introduced the notion of layered treewidth. Loosely speaking, a graph has bounded layered treewidth if it has a tree decomposition and a layering such that each bag of the tree decomposition contains a bounded number of vertices in each layer (defined formally below). This definition is interesting since several natural graph classes, such as planar graphs, that have unbounded treewidth have bounded layered treewidth. Bannister, Devanny, Dujmovi´c,Eppstein, and Wood [1] introduced layered pathwidth, which is analogous to layered treewidth where the tree decomposition is arXiv:1810.08314v2 [math.CO] 4 Jun 2020 required to be a path decomposition. zSchool of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada ([email protected]). Research supported by NSERC and the Ontario Ministry of Research and Innovation. xD´epartement d'Informatique, Universit´eLibre de Bruxelles, Brussels, Belgium ([email protected]). Supported by an ARC grant from the Wallonia-Brussels Federation of Belgium. yDepartment of Computer Science, University of California, Irvine, California, USA ([email protected]). Supported in part by NSF grants CCF-1618301 and CCF-1616248. ∗School of Computer Science, Carleton University, Ottawa, Canada ([email protected]). Research supported by NSERC. {School of Mathematics, Monash University, Melbourne, Australia ([email protected]). Research supported by the Australian Research Council. 1 The purpose of this paper is to characterise the minor-closed graph classes with bounded layered pathwidth. 1.1 Definitions Before continuing, we define the above notions. A tree decomposition of a graph G is a collection (Bx ⊆ V (G): x 2 V (T )) of subsets of V (G) (called bags) indexed by the nodes of a tree T , such that: (i) for every edge uv of G, some bag Bx contains both u and v, and (ii) for every vertex v of G, the set fx 2 V (T ): v 2 Bxg induces a non-empty connected subtree of T . The width of a tree decomposition is the size of the largest bag minus 1. The treewidth of a graph G, denoted by tw(G), is the minimum width of a tree decomposition of G. A path decomposition is a tree decomposition in which the underlying tree is a path. We denote a path decomposition by the corresponding sequence of bags (B1;:::;Bn). The pathwidth of G, denoted by pw(G), is the minimum width of a path decomposition of G. A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. A class of graphs G is minor-closed if for every G 2 G, every minor of G is in G. A layering of a graph G is a partition (V0;V1;:::;Vt) of V (G) such that for every edge vw 2 E(G), if v 2 Vi and w 2 Vj then ji − jj 6 1. Each set Vi is called a layer. For example, for a vertex r of a connected graph G, if Vi is the set of vertices at distance i from r, then (V0;V1;::: ) is a layering of G, called the bfs layering of G starting from r. Dujmovi´cet al. [19] introduced the following definition. The layered width of a tree decomposition (Bx : x 2 V (T )) of a graph G is the minimum integer ` such that, for some layering (V0;V1;:::;Vt) of G, each bag Bx contains at most ` vertices in each layer Vi. The layered treewidth of a graph G, denoted by ltw(G), is the minimum layered width of a tree decomposition of G. Bannister et al. [1] defined the layered pathwidth of a graph G, denoted by lpw(G), to be the minimum layered width of a path decomposition of G. For a graph parameter f (such as treewidth, pathwidth, layered treewidth or layered pathwidth), a graph class G has bounded f is there exists a constant c such that f(G) 6 c for every graph G 2 G. 1.2 Examples and Applications Several interesting graph classes have bounded layered treewidth (despite having unbounded treewidth). For example, Dujmovi´cet al. [19] proved that every planar graph has layered treewidth at most 3, and more generally that every graph with Euler genus g has layered treewidth at most 2g + 3. Note that layered treewidth and layered pathwidth are not minor-closed parameters (unlike treewidth and pathwidth). In fact, several graph classes 2 that contain arbitrarily large clique minors have bounded layered treewidth or bounded layered pathwidth. For example, Dujmovi´c,Eppstein, and Wood [15] proved that every graph that can be drawn on a surface of Euler genus g with at most k crossings per edge has layered treewidth at most 2(2g + 3)(k + 1). Even with g = 0 and k = 1, this family includes graphs with arbitrarily large clique minors. Map graphs have similar behaviour [15]. Bannister et al. [1] identified the following natural graph classes1 that have bounded layered pathwidth (despite having unbounded pathwidth): every squaregraph has layered pathwidth 1; every bipartite outerplanar graph has layered pathwidth 1; every outerplanar graph has layered pathwidth at most 2; every Halin graph has layered pathwidth at most 2; and every unit disc graph with clique number k has layered pathwidth at most 4k. Part of the motivation for studying graphs with bounded layered treewidth or pathwidth is that such graphs have several desirable properties. For example, Norin proved that every p n-vertex graph with layered treewidth k has treewidth less than 2 kn (see [19]). This leads to a very simple proof of the Lipton-Tarjan separator theorem. A standard trick leads to an p upper bound of 11 kn on the pathwidth (see [15]). Another application is to stack layouts (or book embeddings), queue layouts and track layouts. Dujmovi´cet al. [19] proved that every n-vertex graph with layered treewidth k has track- and queue-number O(k log n). This leads to the best known bounds on the track- and queue-number of several natural graph classes2. For graphs with bounded layered pathwidth, the dependence on n can be eliminated: Bannister et al. [1] proved that every graph with layered pathwidth k has track- and queue-number at most 3k. Similarly, Dujmovi´c,Morin, and Yelle [20] proved that every graph with layered pathwidth k has stack-number at most 4k. Graph colouring is another application area for layered treewidth. Esperet and Joret [22] proved that every graph with maximum degree ∆ and Euler genus g is (improperly) 3- colourable with bounded clustering, which means that each monochromatic component has size bounded by some function of ∆ and g. This resolved an old open problem even in the planar case (g = 0). The clustering function proved by Esperet and Joret [22] is roughly O(∆32∆ 2g ). While Esperet and Joret [22] made no effort to reduce this function, their method will not lead to a sub-exponential clustering bound. On the other hand, Liu and Wood [30] proved that every graph with layered treewidth k and maximum degree ∆ is 3-colourable with clustering O(k19∆37), which was recently improved to O(k3∆2) by Dujmovi´c,Esperet, Morin, Walczak, and Wood [16]. In particular, every graph with Euler genus g and maximum degree ∆ is 3-colorable with clustering O(g3∆2). This greatly improves upon the clustering bound of Esperet and Joret [22]. Moreover, the proofs in [16, 30] are relatively simple, avoiding many technicalities that arise when dealing with graph embeddings. These results highlight the utility of layered treewidth as a general tool. 1A squaregraph is a graph that has an embedding in the plane in which each bounded face is a 4-cycle and each vertex either belongs to the unbounded face or has degree at least 4. A Halin graph is a planar graph obtained from a tree T with at least four vertices and with no vertices of degree 2 by adding a cycle through the leaves of T in the clockwise order determined by a plane embedding of T . For a set P of points in the plane, the unit disc graph G of P has vertex set P , where vw 2 E(G) if and only if dist(v; w) 6 1. 2Subsequent to the submission of this paper, improved bounds on the queue-number have been obtained [17]. Still it is open whether graphs of bounded layered treewidth have bounded queue-number. 3 1.3 Characterisations We now turn to the question of characterising those minor-closed classes that have bounded treewidth. The key example is the n×n grid graph, which has treewidth n.